INTRODUCTION. I am trying to understand the proof of the fractional Hardy inequality $$\tag{1} \left\lVert \frac{f(x)}{\lvert x \rvert^s}\right\rVert_{L^2_x(\mathbb R^d)}\le C_{s, d}\lVert f\rVert_{\dot{H}^s(\mathbb R^d)},$$ taken from Lemma A.2 of T.Tao "Nonlinear dispersive equations", CBMS.
The proof relies upon a Littlewood--Paley decomposition $f=\sum_N P_N f$, where $N$ ranges over dyadic integers and $$ \widehat{P_N f}(\xi)=\left( \phi(\xi/N)-\phi(2\xi/N)\right)\hat{f}(\xi), $$ with $\phi$ being a smooth bump function that equals $1$ on the unit ball. The relevance of this decomposition to (1) lies in the formula $$\tag{2} \lVert f\rVert_{\dot{H}^s(\mathbb R^d)}^2=\sum_N N^{2s}\lVert P_N f\rVert_{L^2(\mathbb R^d)}^2.$$
Now, Tao's proof runs across the following lines. First, he decomposes the spatial variable $x$. Precisely, he notes the following bound for the left-hand side of (1): $$\int_{\mathbb R^d}\left\lvert \frac{f(x)}{\lvert x \rvert^s}\right\rvert^2 dx \lesssim \sum_R R^{-2s}\int_{\lvert x \rvert\le R} \lvert f(x)\rvert^2 dx, $$ where $R$ runs over dyadic integers. So far, so good.
QUESTION. At this point, Tao fixes $R$ and focuses on a single integral $\int_{\lvert x \rvert\le R} \lvert f(x)\rvert^2 dx$. He claims that $$\tag{3} \left(\int_{\lvert x \rvert\le R} \lvert f(x)\rvert^2 dx\right)^\frac12 \le \sum_N \left( \int_{\lvert x \rvert \le R} \lvert P_N f(x)\rvert^2 dx\right)^\frac12,$$ where $N$ runs over the dyadic integers. This should "follow from Littlewood--Paley decomposition and the triangle inequality". How to prove (3)?
I am a bit confused because this looks like a spatially localized version of the decomposition $$ \lVert f\rVert_{L^2(\mathbb R^d)}^2\sim \sum_N \lVert P_N f\rVert_{L^2(\mathbb R^d)}^2.$$ Now, the latter is obvious. But how can you infer (3), where you integrate on $\lvert x\rvert\le R$? You are going to have all sorts of mixed terms $\int_{\lvert x \rvert \le R}P_{N_1}f \overline{P_{N_2} f} dx$. As $R\to \infty$, these vanish by essential disjointness of Fourier supports. But here $R$ is fixed.
This is due to peek-a-boo in comments. The standard Littlewood--Paley entails $$ \lvert f \rvert^2=\lim_{\lvert N_0\rvert \to \infty} \left\lvert \sum_{\lvert N\rvert \le \lvert N_0\rvert} P_N f\right\rvert^2, $$ where $N$ and $N_0$ run over the dyadic integers. So by Fatou's lemma, letting $B_R=\{x\in \mathbb R^d\ :\ \lvert x \rvert\le R\}$, $$ \int_{B_R}\lvert f\rvert^2\le \liminf_{\lvert N_0\rvert \to \infty} \int_{B_R} \left\lvert \sum_{\lvert N\rvert\le \lvert N_0\rvert} P_N f\right\rvert^2=\int_{B_R} \left\lvert\sum_{N} P_N f\right\rvert^2,$$ and now (3) follows by taking the square root and applying the triangle inequality.